248] 
EVERY ALGEBRAIC EQUATION HAS A ROOT. 
119 
so that 
Pd = Ar n cos (nd + a), Qd = Ar n sin (:nd + a), 
or the intersections with the cone P = 0 are the n lines given in direction by the 
equation 
nd + a. = (m + 7T, 
and the intersections with the cone Q = 0 are the n lines given in direction by the 
equation 
n6 + a — mir; 
in each of which equations m is any integer number from 0 to n — 1. Hence the 
plane z = 0 meets the cones in two sets of lines succeeding each other alternately, as 
required by the lemma, and the two cones intersect in at least n lines. And it is 
thus shown that the given equation of the wth degree has n roots. 
2, Stone Buildings, W.C., September 26, 1859.